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Palindromic Merlon Primes

Palindromic Merlon Primes (or PMP's for short) are numbers that
are (probable) primes, palindromic in base 10, and consisting of one central digit
(hereby named as a merlon_digit) surrounded by two symmetrical crennelations
with same digits different from the central merlon_digit and finally bordered left
and right by that same central merlon_digit. E.g.

 3223223 31111111111111111311111111111111113

From these examples you will understand the naming of these kind of palindromic primes Visualising merlons

Some links where PMP's are discussed ¬
Liczby pierwsze o szczególnym rozmieszczeniu cyfr by Andrzej Nowicki
Translated in Dutch “Priemgetallen met een speciale rangschikking van cijfers”
Translated in English “Prime numbers with a special arrangement of digits”
1221221 from the site 'Darkbyte'
3223223 from Prime Curios!

In case one should discover more sources I will be most happy
to add them to the list. Just let me know.

PMP's sorted by length

Some combinations can never produce primes since
these are always divisible by 3.
1(0)w1(0)w1
1(3)w1(3)w1
1(6)w1(6)w1
1(9)w1(9)w1
3(0)w3(0)w3
3(6)w3(6)w3
3(9)w3(9)w3
7(0)w7(0)w7
7(3)w7(3)w7
7(6)w7(6)w7
7(9)w7(9)w7
9(0)w9(0)w9
9(3)w9(3)w9
9(6)w9(6)w9

PMP Factorization Projects

( n = 2 * w + 3 )

 PMP (Palindromic Merlon Primes) reference files.Members can be prime.All files maintained by Patrick De Geest. 1(2)w1(2)w1 = 2*(10n–1)/9 – 10n-1 – 10(n-1)/2 – 1 facpmp121.htm Free to factor    16 remaining 1(4)w1(4)w1 = 4*(10n–1)/9 – 3*10n-1 – 3*10(n-1)/2 – 3 facpmp141.htm Free to factor    12 remaining 1(5)w1(5)w1 = 5*(10n–1)/9 – 4*10n-1 – 4*10(n-1)/2 – 4 facpmp151.htm Free to factor    10 remaining 1(7)w1(7)w1 = 7*(10n–1)/9 – 6*10n-1 – 6*10(n-1)/2 – 6 facpmp171.htm Free to factor    13 remaining 1(8)w1(8)w1 = 8*(10n–1)/9 – 7*10n-1 – 7*10(n-1)/2 – 7 facpmp181.htm Free to factor    20 remaining 3(1)w3(1)w3 = 3*(10n–1)/9 + 2*10n-1 + 2*10(n-1)/2 + 2 facpmp313.htm Free to factor    14 remaining 3(2)w3(2)w3 = 2*(10n–1)/9 + 10n-1 + 10(n-1)/2 + 1 facpmp323.htm Free to factor    12 remaining 3(4)w3(4)w3 = 4*(10n–1)/9 – 10n-1 – 10(n-1)/2 – 1 facpmp343.htm Free to factor    13 remaining 3(5)w3(5)w3 = 5*(10n–1)/9 – 2*10n-1 – 2*10(n-1)/2 – 2 facpmp353.htm Free to factor    18 remaining 3(7)w3(7)w3 = 7*(10n–1)/9 – 4*10n-1 – 4*10(n-1)/2 – 4 facpmp373.htm Free to factor    11 remaining 3(8)w3(8)w3 = 8*(10n–1)/9 – 5*10n-1 – 5*10(n-1)/2 – 5 facpmp383.htm Free to factor    14 remaining 7(1)w7(1)w7 = (10n–1)/9 + 6*10n-1 + 6*10(n-1)/2 + 6 facpmp717.htm Free to factor    14 remaining 7(2)w7(2)w7 = 2*(10n–1)/9 + 5*10n-1 + 5*10(n-1)/2 + 5 facpmp727.htm Free to factor    14 remaining 7(4)w7(4)w7 = 4*(10n–1)/9 + 3*10n-1 + 3*10(n-1)/2 + 3 facpmp747.htm Free to factor    16 remaining 7(5)w7(5)w7 = 5*(10n–1)/9 + 2*10n-1 + 2*10(n-1)/2 + 2 facpmp757.htm Free to factor    19 remaining 7(8)w7(8)w7 = 8*(10n–1)/9 – 10n-1 – 10(n-1)/2 + 2 facpmp787.htm Free to factor    16 remaining 9(1)w9(1)w9 = (10n–1)/9 + 8*10n-1 + 8*10(n-1)/2 + 8 facpmp919.htm Free to factor    12 remaining 9(2)w9(2)w9 = 2*(10n–1)/9 + 7*10n-1 + 7*10(n-1)/2 + 7 facpmp929.htm Free to factor    13 remaining 9(4)w9(4)w9 = 4*(10n–1)/9 + 5*10n-1 + 5*10(n-1)/2 + 5 facpmp949.htm Free to factor    17 remaining 9(5)w9(5)w9 = 5*(10n–1)/9 + 4*10n-1 + 4*10(n-1)/2 + 4 facpmp959.htm Free to factor    19 remaining 9(7)w9(7)w9 = 7*(10n–1)/9 + 2*10n-1 + 2*10(n-1)/2 + 2 facpmp979.htm Free to factor    13 remaining 9(8)w9(8)w9 = 8*(10n–1)/9 + 10n-1 + 10(n-1)/2 + 1 facpmp989.htm Free to factor    20 remaining
 PMP (Palindromic Merlon Primes) reference files. Members (n>1) are always composite.All files maintained by Patrick De Geest  [ Factorization Files Under Construction ] 1(0)w1(0)w1 = 10n + 10n/2 + 1 → (n even) facpmp101.htm All factored (w ⩽ 100) 1(3)w1(3)w1 = 3*(10n–1)/9 – 2.10n-1 – 2.10(n-1)/2 – 2 facpmp131.htm Free to factor    12 remaining 1(6)w1(6)w1 = 6*(10n–1)/9 – 5.10n-1 – 5.10(n-1)/2 – 5 facpmp161.htm Free to factor    11 remaining 1(9)w1(9)w1 = 9*(10n–1)/9 – 8.10n-1 – 8.10(n-1)/2 – 8 facpmp191.htm Free to factor    5 remaining 2(1)w2(1)w2 = (10n–1)/9 + 10n-1 + 10(n-1)/2 + 1 facpmp212.htm Free to factor    6 remaining 2(3)w2(3)w2 = 3*(10n–1)/9 – 10n-1 – 10(n-1)/2 – 1 facpmp232.htm All factored (w ⩽ 100) 2(5)w2(5)w2 = 5*(10n–1)/9 – 3*10n-1 – 3*10(n-1)/2 – 3 facpmp252.htm Free to factor    9 remaining 2(7)w2(7)w2 = 7*(10n–1)/9 – 5*10n-1 – 5*10(n-1)/2 – 5 facpmp272.htm All factored (w ⩽ 100) 2(9)w2(9)w2 = 9*(10n–1)/9 – 7*10n-1 – 7*10(n-1)/2 – 7 facpmp292.htm Free to factor    13 remaining 4(1)w4(1)w4 = (10n–1)/9 + 3*10n-1 + 3*10(n-1)/2 + 3 facpmp414.htm Free to factor    12 remaining 4(3)w4(3)w4 = 3*(10n–1)/9 + 10n-1 + 10(n-1)/2 + 1 facpmp434.htm Free to factor    11 remaining 4(5)w4(5)w4 = 5*(10n–1)/9 – 10n-1 – 10(n-1)/2 – 1 facpmp454.htm Free to factor    12 remaining 4(7)w4(7)w4 = 7*(10n–1)/9 – 3*10n-1 – 3*10(n-1)/2 – 3 facpmp474.htm Free to factor    10 remaining 4(9)w4(9)w4 = 9*(10n–1)/9 – 5*10n-1 – 5*10(n-1)/2 – 5 facpmp494.htm Free to factor    10 remaining 5(1)w5(1)w5 = (10n–1)/9 + 4*10n-1 + 4*10(n-1)/2 + 4 facpmp515.htm Free to factor    13 remaining 5(2)w5(2)w5 = 2*(10n–1)/9 + 3*10n-1 + 3*10(n-1)/2 + 3 facpmp525.htm Free to factor    17 remaining 5(3)w5(3)w5 = 3*(10n–1)/9 + 2*10n-1 + 2*10(n-1)/2 + 2 facpmp535.htm Free to factor    13 remaining 5(4)w5(4)w5 = 4*(10n–1)/9 + 10n-1 + 10(n-1)/2 + 1 facpmp545.htm Free to factor    12 remaining 5(6)w5(6)w5 = 6*(10n–1)/9 – 10n-1 – 10(n-1)/2 – 1 facpmp565.htm Free to factor    9 remaining 5(7)w5(7)w5 = 7*(10n–1)/9 – 2*10n-1 – 2*10(n-1)/2 – 2 facpmp575.htm Free to factor    7 remaining 5(8)w5(8)w5 = 8*(10n–1)/9 – 3*10n-1 – 3*10(n-1)/2 – 3 facpmp585.htm 5(9)w5(9)w5 = 9*(10n–1)/9 – 4*10n-1 – 4*10(n-1)/2 – 4 facpmp595.htm 6(1)w6(1)w6 = (10n–1)/9 + 5*10n-1 + 5*10(n-1)/2 + 5 facpmp616.htm 6(5)w6(5)w6 = 5*(10n–1)/9 + 10n-1 + 10(n-1)/2 + 1 facpmp656.htm 6(7)w6(7)w6 = 7*(10n–1)/9 – 10n-1 – 10(n-1)/2 – 1 facpmp676.htm 7(3)w7(3)w7 = 3*(10n–1)/9 + 4*10n-1 + 4*10(n-1)/2 + 4 facpmp737.htm 7(6)w7(6)w7 = 6*(10n–1)/9 + 10n-1 + 10(n-1)/2 + 1 facpmp767.htm 7(9)w7(9)w7 = 9*(10n–1)/9 – 2*10n-1 – 2*10(n-1)/2 – 2 facpmp797.htm 8(1)w8(1)w8 = (10n–1)/9 + 7*10n-1 + 7*10(n-1)/2 + 7 facpmp818.htm 8(3)w8(3)w8 = 3*(10n–1)/9 + 5*10n-1 + 5*10(n-1)/2 + 5 facpmp838.htm 8(5)w8(5)w8 = 5*(10n–1)/9 + 3*10n-1 + 3*10(n-1)/2 + 3 facpmp858.htm 8(7)w8(7)w8 = 7*(10n–1)/9 + 10n-1 + 10(n-1)/2 + 1 facpmp878.htm 8(9)w8(9)w8 = 9*(10n–1)/9 – 10n-1 – 10(n-1)/2 – 1 facpmp898.htm

[ February 4, 2023 ]
Chen Xinyao informs me that
https://www.alpertron.com.ar/MODFERM.HTM is the factorization of the PMPs 1(0^^w)1(0^^w)1 in base 2.

Following condition must be imposed that gcd(A,B) = 1 (in Factorization of ABB...BBABB...BBA), i.e. A and B are coprime, since if A and B have a common factor > 1, then we can divide this
factor from the number, e.g. factor 6999...9996999...9996 is equivalent to factor 2333...3332333...3332.

[ April 18, 2023 ]
Chen Xinyao informs me that
some PMPs have algebraic factors: (n=w+1)

2(3^^w)2(3^^w)2 3*(10^(2*n+1)-1)/9-10^(2*n)-10^n-1 = ((2^n*5^n-1)*(7*2^n*5^n+4))/3 = (1^^n) * (7(0^^(n-1)4) * 3
2(7^^w)2(7^^w)2 7*(10^(2*n+1)-1)/9-5*10^(2*n)-5*10^n-5 = ((2^n*5^(n+1)-13)*(2^n*5^(n+1)+4))/9 = 4(9^^(n-2))87 * 5(0^^(n-1))4 / 9

I have checked all combinations of PMP, only 2333...3332333...3332 and 2777...7772777...7772 have algebraic factorization.

Thus link to Kamada's pages factorization of (1^^n), factorization of 7(0^^n)4, factorization of 2*5(0^^n)4
Unfortunately, Kamada's page has no factorization of 4(9^^n)87 or any related numbers since it only has
(x^^n), (x^^n)y, x(y^^n), (x^^n)yx, xy(x^^n), x(y^^n)x, x(y^^n)z, and (x^^n)y(x^^n).

Also 5*10^n-13 (or 4(9^^(n-2))87) is already fully factored for all n<137,
see http://factordb.com/index.php?query=5*10%5En-13&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show

[ February 3, 2024 ]
Patrick De Geest is always on the search for more or less beautiful patterns, curiosities and observations.

 PATTERNS in the expansions of the Palindromic Merlon Numbers of type PMP545 (4*(10^(n–1))/9 + 10^(n-1) + 10^((n-1)/2) + 1) divided by a combination of its smallest factors 3, 5 & 37 with multiplicity. Divided by 3 E.g. n = 33 Alternations of the digits 1, 4 and 8 except for the last digit which is a 5. 181481481481481484814814814814815 Divided by 5 E.g. n = 33 Strings of 8's sandwiched between 10, 90 and 9. Their lengths are defined by (n-5)/2 so in this example it is 14.Note that 10909 is a prime. 10{88888888888888}90{88888888888888}9 Divided by 3 * 3 = 9 E.g. n = 33 The last ten digits constitute a pandigital number 4938271605 whereby the digits from 0 to 9 are intertwinedand ascending from the right to the left →   4938271605 6049382716049382827160_4938271605 Divided by 3 * 5 = 15 E.g. n = 33 A string made of the digits 2, 6 and 9 squeezed between two 3's. Note it is not a palindrome. 3_629629629629629696296296296296_3 Divided by 3 * 3 * 3 = 27 E.g. n = 87 Here also lurks a pandigital number at the end of the decimal expansion but now divided in five dispersed duo's.…90534979423868312757201646090535 …90534979423868312757201646090535 Divided by 3 * 3 * 5 = 45 E.g. n = 195 All the ten digits from 0 to 9 appear at the start and the end of the decimal expansion.Adding up 1209876543 + 0987654321 gives 2197530864 which is also a pandigital number.And subtracting both numbers gives surprisingly 222222222 or a repdigit. 1209876543_20987654320…098765432098765432_0987654321 Divided by 3 * 37 = 111 E.g. n = 27 Alternations of the digits 0, 4 and 9 except for the last digit which is a 5. 4904904904904994994994995 Divided by 3 * 3 * 3 * 5 = 135 E.g. n = 141 Nothing spectacular found except that digits 4, 5, 6, 7, 8, 9 often appear as doubles.…0329218106 99 5 88 4 77 3 66 2 55 1 44 0329218107Intersperced between these doubles we observe a descending sequence from digit 6 down to 0. 40329218106…0329218106995884773662551440329218107 Divided by 5 * 37 = 185 E.g. n = 51 Alternations of the digits 2, 4 and 9 for the left partand 6 and 9 for the right side except for the last digit which is a 7. 2942942942942942942942942996996996996996996996997 Divided by 3 * 3 * 37 = 333 E.g. n = 51 Digits 2 and 7 do not appear in the decimal expansion. 1634968301634968301634968331664998331664998331665 Divided by 3 * 5 * 37 = 555 E.g. n = 135 Alternations of only the digits 0, 8 and 9. 98098098098098098098…98998998998998998999 Divided by 3 * 3 * 3 * 37 = 999 E.g. n = 195 Occurence of triplets like 555, 666, 777, 888, 999. 555 happens only once at the very end of the decimal expansion. …999443888332777221666110554999443888332777221666110555 Divided by 3 * 3 * 5 * 37 = 1665 E.g. n = 141 Ends with a triplet of the digits 3, 6 and 9. …999666332999666332999666332_999666333 Divided by 3 * 3 * 3 * 3 * 37 = 2997 E.g. n = 141 I see a few triplets but nothing spectacular... Do you see more ? 181663144626107589070552033514996477959440922403885366848329811292774 259073888703518333147962777592407222036851666481296110925740555370185 Divided by 3 * 3 * 3 * 5 * 37 = 4995 E.g. n = 141 Ends with a triplet of all the digits from 999 down to 111 separated from the rest by one zero.…10999888777666555444333222110{999}{888}{777}{666}{555}{444}{333}{222}{111} …10999888777666555444333222110999888777666555444333222111 Divided by 3 * 3 * 3 * 3 * 5 * 37 = 14985 E.g. n = 141 Again, this is a case with triplets occurring for digits 9 down to 7 and 1 to 3 upwards.Inbetweeners 4,5,6 and 7 gather even together in quartets.36332628925221517814110406702{999}295591{888}184480{777}073369665962258554851814{7777}40703{6666}29592{5555}18481{4444}07370{333}296259{222}185148{111}074037 363326289252215178141104067029992955918881844807770733696659622585548 51814777740703666629592555518481444407370333296259222185148111074037

Can you reveal more intricate patterns? If so, just let me know and I'll add them also.

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The “PMP” Table

 ¬ PDG Nov 12 2011 PRIME View PDG The reference table forPalindromic Merlon Primes This collection is complete forprobable primes up to 50000digits and for provenprimes up to  3500  digits. `PDG = Patrick De Geest` PMP Formulablue exp = # of digitsAccolades = prime exp Who When Status OutputLogs n ⩾ 50011 (PDG, September 27, 2022) 1(2)21(2)21 2*(10{7}–1)/9 – 106 – 103 – 1 1(2)41(2)41 2*(10{11}–1)/9 – 1010 – 105 – 1 1(2)111(2)111 2*(1025–1)/9 – 1024 – 1012 – 1 1(2)23921(2)23921 2*(10{4787}–1)/9 – 104786 – 102393 – 1 1(2)188141(2)188141 2*(1037631–1)/9 – 1037630 – 1018815 – 1 n ⩾ 50191 (PDG, September 28, 2022) 1(4)441(4)441 4*(1091–1)/9 – 3*1090 – 3*1045 – 3 1(4)641(4)641 4*(10{131}–1)/9 – 3*10130 – 3*1065 – 3 1(4)91491(4)91491 4*(10{18301}–1)/9 – 3*1018300 – 3*109150 – 3 1(4)228261(4)228261 4*(1045655–1)/9 – 3*1045654 – 3*1022827 – 3 n ⩾ 53947 (PDG, September 28, 2022) 1(5)21(5)21 5*(10{7}–1)/9 – 4*106 – 4*103 – 4 1(5)81(5)81 5*(10{19}–1)/9 – 4*1018 – 4*109 – 4 1(5)321(5)321 5*(10{67}–1)/9 – 4*1066 – 4*1033 – 4 1(5)1281(5)1281 5*(10259–1)/9 – 4*10258 – 4*10129 – 4 1(5)4941(5)4941 5*(10{991}–1)/9 – 4*10990 – 4*10495 – 4 1(5)42611(5)42611 5*(108525–1)/9 – 4*108524 – 4*104262 – 4 1(5)137651(5)137651 5*(1027533–1)/9 – 4*1027532 – 4*1013766 – 4 n ⩾ 56095 (PDG, September 29, 2022) 1(7)41(7)41 7*(10{11}–1)/9 – 6*1010 – 6*105 – 6 1(7)221(7)221 7*(10{47}–1)/9 – 6*1046 – 6*1023 – 6 1(7)3161(7)3161 7*(10635–1)/9 – 6*10634 – 6*10317 – 6 1(7)4421(7)4421 7*(10{887}–1)/9 – 6*10886 – 6*10443 – 6 n ⩾ 58103 (PDG, September 29, 2022) 1(8)11(8)11 8*(10{5}–1)/9 – 7*104 – 7*102 – 7 1(8)21(8)21 8*(10{7}–1)/9 – 7*106 – 7*103 – 7 1(8)1451(8)1451 8*(10{293}–1)/9 – 7*10292 – 7*10146 – 7 1(8)2541(8)2541 8*(10511–1)/9 – 7*10510 – 7*10255 – 7 1(8)16271(8)16271 8*(10{3257}–1)/9 – 7*103256 – 7*101628 – 7 1(8)18131(8)18131 8*(103629–1)/9 – 7*103628 – 7*101814 – 7 n ⩾ 53315 (PDG, September 30, 2022) 3(1)163(1)163 (1035–1)/9 + 2*1034 + 2*1017 + 2 3(1)433(1)433 (10{89}–1)/9 + 2*1088 + 2*1044 + 2 3(1)863(1)863 (10175–1)/9 + 2*10174 + 2*1087 + 2 3(1)11533(1)11533 (10{2309}–1)/9 + 2*102308 + 2*101154 + 2 n ⩾ 50821 (PDG, October 1, 2022) 3(2)13(2)13 2*(10{5}–1)/9 + 104 + 102 + 1 3(2)23(2)23 2*(10{7}–1)/9 + 106 + 103 + 1 3(2)53(2)53 2*(10{13}–1)/9 + 1012 + 106 + 1 3(2)19033(2)19033 2*(10{3809}–1)/9 + 103808 + 101904 + 1 3(2)29533(2)29533 2*(105909–1)/9 + 105908 + 102954 + 1 3(2)34133(2)34133 2*(10{6829}–1)/9 + 106828 + 103414 + 1 n ⩾ 50189 (PDG, October 2, 2022) 3(4)23(4)23 4*(10{7}–1)/9 – 106 – 103 – 1 3(4)43(4)43 4*(10{11}–1)/9 – 1010 – 105 – 1 3(4)73(4)73 4*(10{17}–1)/9 – 1016 – 108 – 1 3(4)223(4)223 4*(10{47}–1)/9 – 1046 – 1023 – 1 3(4)263(4)263 4*(1055–1)/9 – 1054 – 1027 – 1 3(4)1823(4)1823 4*(10{367}–1)/9 – 10366 – 10183 – 1 3(4)2053(4)2053 4*(10413–1)/9 – 10412 – 10206 – 1 3(4)4763(4)4763 4*(10955–1)/9 – 10954 – 10477 – 1 3(4)13193(4)13193 4*(102641–1)/9 – 102640 – 101320 – 1 3(4)127423(4)127423 4*(1025487–1)/9 – 1025486 – 1012743 – 1 3(4)172433(4)172433 4*(1034489–1)/9 – 1034488 – 1017244 – 1 n ⩾ 50657 (PDG, October 2, 2022) 3(5)13(5)13 5*(10{5}–1)/9 – 2*104 – 2*102 – 2 3(5)23(5)23 5*(10{7}–1)/9 – 2*106 – 2*103 – 2 3(5)173(5)173 5*(10{37}–1)/9 – 2*1036 – 2*1018 – 2 3(5)203(5)203 5*(10{43}–1)/9 – 2*1042 – 2*1021 – 2 3(5)263(5)263 5*(1055–1)/9 – 2*1054 – 2*1027 – 2 3(5)1573(5)1573 5*(10{317}–1)/9 – 2*10316 – 2*10158 – 2 3(5)6143(5)6143 5*(10{1231}–1)/9 – 2*101230 – 2*10615 – 2 3(5)8333(5)8333 5*(10{1669}–1)/9 – 2*101668 – 2*10834 – 2 3(5)33613(5)33613 5*(106725–1)/9 – 2*106724 – 2*103362 – 2 3(5)36313(5)36313 5*(107265–1)/9 – 2*107264 – 2*103632 – 2 3(5)38443(5)38443 5*(10{7691}–1)/9 – 2*107690 – 2*103845 – 2 n ⩾ 55429 (PDG, October 3, 2022) 3(7)23(7)23 7*(10{7}–1)/9 – 4*106 – 4*103 – 4 3(7)43(7)43 7*(10{11}–1)/9 – 4*1010 – 4*105 – 4 3(7)473(7)473 7*(10{97}–1)/9 – 4*1096 – 4*1048 – 4 3(7)593(7)593 7*(10121–1)/9 – 4*10120 – 4*1060 – 4 3(7)703(7)703 7*(10143–1)/9 – 4*10142 – 4*1071 – 4 3(7)1223(7)1223 7*(10247–1)/9 – 4*10246 – 4*10123 – 4 3(7)1283(7)1283 7*(10259–1)/9 – 4*10258 – 4*10129 – 4 3(7)60943(7)60943 7*(1012191–1)/9 – 4*1012190 – 4*106095 – 4 3(7)85243(7)85243 7*(1017051–1)/9 – 4*1017050 – 4*108525 – 4 3(7)189893(7)189893 7*(1037981–1)/9 – 4*1037980 – 4*1018990 – 4 n ⩾ 50417 (PDG, October 4, 2022) 3(8)83(8)83 8*(10{19}–1)/9 – 5*1018 – 5*109 – 5 3(8)103(8)103 8*(10{23}–1)/9 – 5*1022 – 5*1011 – 5 3(8)143(8)143 8*(10{31}–1)/9 – 5*1030 – 5*1015 – 5 3(8)673(8)673 8*(10{137}–1)/9 – 5*10136 – 5*1068 – 5 3(8)3643(8)3643 8*(10731–1)/9 – 5*10730 – 5*10365 – 5 3(8)5783(8)5783 8*(101159–1)/9 – 5*101158 – 5*10579 – 5 3(8)8483(8)8483 8*(10{1699}–1)/9 – 5*101698 – 5*10849 – 5 3(8)30763(8)30763 8*(106155–1)/9 – 5*106154 – 5*103077 – 5 3(8)78403(8)78403 8*(10{15683}–1)/9 – 5*1015682 – 5*107841 – 5 3(8)142063(8)142063 8*(1028415–1)/9 – 5*1028414 – 5*1014207 – 5 3(8)193993(8)193993 8*(1038801–1)/9 – 5*1038800 – 5*1019400 – 5 3(8)233963(8)233963 8*(1046795–1)/9 – 5*1046794 – 5*1023397 – 5 n ⩾ 50681 (PDG, October 4, 2022) 7(1)557(1)557 (10{113}–1)/9 + 6*10112 + 6*1056 + 6 n ⩾ 51193 (PDG, October 5, 2022) 7(2)17(2)17 2*(10{5}–1)/9 + 5*104 + 5*102 + 5 7(2)47(2)47 2*(10{11}–1)/9 + 5*1010 + 5*105 + 5 7(2)77(2)77 2*(10{17}–1)/9 + 5*1016 + 5*108 + 5 7(2)227(2)227 2*(10{47}–1)/9 + 5*1046 + 5*1023 + 5 7(2)297(2)297 2*(10{61}–1)/9 + 5*1060 + 5*1030 + 5 7(2)497(2)497 2*(10{101}–1)/9 + 5*10100 + 5*1050 + 5 7(2)737(2)737 2*(10{149}–1)/9 + 5*10148 + 5*1074 + 5 7(2)837(2)837 2*(10169–1)/9 + 5*10168 + 5*1084 + 5 7(2)1187(2)1187 2*(10{239}–1)/9 + 5*10238 + 5*10119 + 5 7(2)2417(2)2417 2*(10485–1)/9 + 5*10484 + 5*10242 + 5 n ⩾ 51035 (PDG, October 5, 2022) 7(4)17(4)17 4*(10{5}–1)/9 + 3*104 + 3*102 + 3 7(4)1217(4)1217 4*(10245–1)/9 + 3*10244 + 3*10122 + 3 7(4)5207(4)5207 4*(101043–1)/9 + 3*101042 + 3*10521 + 3 7(4)12647(4)12647 4*(10{2531}–1)/9 + 3*102530 + 3*101265 + 3 7(4)17807(4)17807 4*(103563–1)/9 + 3*103562 + 3*101781 + 3 n ⩾ 56255 (PDG, October 5, 2022) 7(5)262727(5)262727 5*(1052547–1)/9 + 2*1052546 + 2*1026273 + 2 n ⩾ 53665 (PDG, October 6, 2022) 7(8)17(8)17 8*(10{5}–1)/9 – 104 – 102 – 1 7(8)47(8)47 8*(10{11}–1)/9 – 1010 – 105 – 1 7(8)1277(8)1277 8*(10{257}–1)/9 – 10256 – 10128 – 1 7(8)3297(8)3297 8*(10{661}–1)/9 – 10660 – 10330 – 1 7(8)8037(8)8037 8*(10{1609}–1)/9 – 101608 – 10804 – 1 7(8)18407(8)18407 8*(103683–1)/9 – 103682 – 101841 – 1 n ⩾ 50395 (PDG, October 7, 2022) 9(1)49(1)49 (10{11}–1)/9 + 8*1010 + 8*105 + 8 9(1)79(1)79 (10{17}–1)/9 + 8*1016 + 8*108 + 8 9(1)299(1)299 (10{61}–1)/9 + 8*1060 + 8*1030 + 8 9(1)469(1)469 (1095–1)/9 + 8*1094 + 8*1047 + 8 9(1)589(1)589 (10119–1)/9 + 8*10118 + 8*1059 + 8 9(1)689(1)689 (10{139}–1)/9 + 8*10138 + 8*1069 + 8 9(1)839(1)839 (10169–1)/9 + 8*10168 + 8*1084 + 8 9(1)9559(1)9559 (10{1913}–1)/9 + 8*101912 + 8*10956 + 8 9(1)11609(1)11609 (102323–1)/9 + 8*102322 + 8*101161 + 8 9(1)55049(1)55049 (10{11011}–1)/9 + 8*1011010 + 8*105505 + 8 9(1)62689(1)62689 (10{12539}–1)/9 + 8*1012538 + 8*106269 + 8 9(1)92909(1)92909 (10{18583}–1)/9 + 8*1018582 + 8*109291 + 8 9(1)217669(1)217669 (1043535–1)/9 + 8*1043534 + 8*1021767 + 8 n ⩾ 52841 (PDG, October 7, 2022) 9(2)49(2)49 2*(10{11}–1)/9 + 7*1010 + 7*105 + 7 9(2)89(2)89 2*(10{19}–1)/9 + 7*1018 + 7*109 + 7 9(2)269(2)269 2*(1055–1)/9 + 7*1054 + 7*1027 + 7 9(2)2029(2)2029 2*(10407–1)/9 + 7*10406 + 7*10203 + 7 9(2)20689(2)20689 2*(10{4139}–1)/9 + 7*104138 + 7*102069 + 7 9(2)63749(2)63749 2*(1012751–1)/9 + 7*1012750 + 7*106375 + 7 n ⩾ 50963 (PDG, October 8, 2022) 9(4)19(4)19 4*(10{5}–1)/9 + 5*104 + 5*102 + 5 9(4)49(4)49 4*(10{11}–1)/9 + 5*1010 + 5*105 + 5 9(4)79(4)79 4*(10{17}–1)/9 + 5*1016 + 5*108 + 5 9(4)209(4)209 4*(10{43}–1)/9 + 5*1042 + 5*1021 + 5 9(4)5099(4)5099 4*(10{1021}–1)/9 + 5*101020 + 5*10510 + 5 n ⩾ 50461 (PDG, October 8, 2022) 9(5)19(5)19 5*(10{5}–1)/9 + 4*104 + 4*102 + 4 9(5)389(5)389 5*(10{79}–1)/9 + 4*1078 + 4*1039 + 4 9(5)1739(5)1739 5*(10{349}–1)/9 + 4*10348 + 4*10174 + 4 9(5)14939(5)14939 5*(102989–1)/9 + 4*102988 + 4*101494 + 4 9(5)229909(5)229919 5*(1045983–1)/9 + 4*1045982 + 4*1022991 + 4 n ⩾ 56783 (PDG, October 8, 2022) 9(7)29(7)29 7*(10{7}–1)/9 + 2*106 + 2*103 + 2 9(7)409(7)409 7*(10{83}–1)/9 + 2*1082 + 2*1041 + 2 9(7)2989(7)2989 7*(10{599}–1)/9 + 2*10598 + 2*10299 + 2 n ⩾ 52507 (PDG, October 9, 2022) 9(8)29(8)29 8*(10{7}–1)/9 + 106 + 103 + 1 9(8)49(8)49 8*(10{11}–1)/9 + 1010 + 105 + 1 9(8)89(8)89 8*(10{19}–1)/9 + 1018 + 109 + 1 9(8)149(8)149 8*(10{31}–1)/9 + 1030 + 1015 + 1 9(8)329(8)329 8*(10{67}–1)/9 + 1066 + 1033 + 1 9(8)569(8)569 8*(10115–1)/9 + 10114 + 1057 + 1 9(8)3829(8)3829 8*(10767–1)/9 + 10766 + 10383 + 1

Sources Revealed

 Neil Sloane's “Integer Sequences” Encyclopedia can be consulted online : Neil Sloane's Integer Sequences Various numbers, primes and palindromic primes are categorised as follows : %N Merlon numbers. Start is identical to sequence A?????? %N Palindromic merlon primes. under A?????? %N Palindromic merlon primes exist for digitlengths a(n). under A?????? Click here to view some of the author's [P. De Geest] entries to the table. Click here to view some entries to the table about palindromes.

Prime Curios! - site maintained by G. L. Honaker Jr. and Chris Caldwell
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All probable primes above 10000 digits are also
submitted to the PRP TOP records table maintained by Henri & Renaud Lifchitz.

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